72 ON THE PROPAGATION OF WAVES [360
transition should move together — equivalent to the ordinary law of refraction. In the usual optical notation, if V be the velocity of propagation and 6 the angle of incidence,
c = 27rF/X, b = (2-7T/\) sin 0, a = (2w/\) cos 0, ...... (2)
where V/\, \~l sin 6 are the same in all the strata. On the other hand a is variable and is connected with the direction of propagation within the stratum by the relation
(3)
The a's are thus known in terms of the original angle of incidence and of the various refractive indices.
Since the factor ei (ct+l>y} runs through all our expressions, we may regard it as understood and write simply
......................... (5)
03 = Ase-ia*{x~x*] + Bseia* (x~x*\ ......................... (6)
(V)
In the problem of reflection we are to make Bm = 0, and (if we please) Am — !•
We have now to consider the boundary conditions which hold at the surfaces of transition. In the case of sound travelling through gas, where 0 is taken to represent the velocity-potential, these conditions are the continuity of d^/dx and of cr0; where a is the density. Whether the multiplier attaches to the dependent variable itself or to its derivative is of no particular significance. For example, if we take a new dependent variable ^r, equal to <r0, the above conditions are equivalent to the continuity of ^r and of a^d^/dsc. Nor should we really gain generality by introducing a multiplier in both places. We may therefore for the present confine ourselves to the acoustical form, knowing that the results will admit of interpretation in numerous other cases.
At the first transition x = x^ the boundary conditions give
ck(B,-A,}^=a,(Bz-Az}i ^(A + A) = <r2(-fi2 -M2) ........ (8)
If we stop here, we have the simple case of the juxtaposition of two media both of infinite depth. Supposing B2 = 0, we get
-Bl _ O-g/O-! — OzfOj _ CTg/QT! — COt 02/COt 01
A! o-g/o-j + tta/Oj o-2/a1 + Cot #2/COt #1 ...................
For a further discussion of (9) reference may be made to Theory of Sound (loc. cit). In the case of the simple gases the compressibilities are made a year ago gave promising results. Ten lantern-slides were prepared from a portrait negative. The exposure (to gas-light) was for about 3 seconds through the negative and for 30 seconds bare, i.e. with negative removed, and the development was rather light. On single plates the picture was but just visible. Some rough photometry indicated that each plate transmitted about one-third of the incident light. In carrying out the exposures suitable stops, cemented to the negative, must be provided to guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.